ASICs are not reprogrammable. And they're expensive to build, but the NSA apparently has the resources to build new ones at semi-regular intervals, so I don't know if switching algorithms would really slow them down that much.
I think they could probably do significantly faster than that since these would presumably be a relatively low volume, single customer application, and wouldn't need the extensive testing cycle required for mass production ASICs. Which is to say it doesn't take a year from delivering a mask set to a fab to getting packaged parts.
I think grandparent was suggesting custom encryption algorithm (potentially less secure) to prevent use of a brute force approach that's custom made for a specific algorithm.
I thought I heard some presentation about superencipherment with AES + some cipher (maybe derived from AES) with per-session randomized S-boxes. You would include the S-boxes in the message as a sort of salt.
The goal of this is that if there is a cryptanalytic attack that gives brute force a 2⁴⁰ speedup and hence attackers build custom hardware to implement it, their hardware is still not effective against the other cipher and they have to do something different (maybe hard to characterize how difficult the different thing would be anticipated to be).
Don't ever do this. The interaction of different crypto algorithms can be counterintuitive (in the sense that you won't gain what you think you're gaining), it's extraordinarily expensive in ways unbalanced against defenders, and it introduces new implementation errors that can weaken the whole system.
If you're paranoid about AES, use Chapoly (or just use Chapoly because it's in many ways more convenient to use than AES). Don't build elaborate cascades (all cascades qualify), and don't randomize S-boxes.
It is hard enough getting these systems right when you're playing exactly by the book. If you're designing a cryptosystem and you're not a professional cryptographer, the list of things you should be worried about getting wrong is very long and very scary.
By "don't build elaborate cascades (all cascades qualify)", do you mean "no cascade construction is preferable to any other", or "every cascade is too elaborate to be worthwhile"? (The second implies the first, but the first doesn't imply the second.)